The Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing

Transcription

1 he Generalized Exreme Value (GEV) Disribuion, Implied ail Index and Opion Pricing Sheri Markose and Amadeo Alenorn his version: 6 December 200 Forhcoming Spring 20 in he Journal of Derivaives Absrac Crisis evens such as he 987 sock marke crash, he Asian Crisis and he collapse of Lehman Brohers have radically changed he view ha exreme evens in financial markes have negligible probabiliy. his aricle argues ha he use of he Generalized Exreme Value (GEV) disribuion o model he implied Risk Neural Densiy (RND) funcion provides a flexible framework ha capures he negaive skewness and excess kurosis of reurns, and also delivers he marke implied ail index. We obain an original analyical closed form soluion for he arrison and Pliska [98] no arbirage equilibrium price for he European opion in he case of GEV asse reurns. he GEV based opion pricing model successfully removes he in-sample pricing bias of he Black-Scholes model, and also shows greaer ou of sample pricing accuracy, while requiring he esimaion of only wo parameers. We explain how he implied ail index is efficacious a modelling he fa ailed behaviour and negaive skewness of he implied RND funcions, paricularly around crisis evens. - -

2 he las wo decades have been marked by crisis evens in financial markes. hese include he 987 sock marke crash, he Asian Crisis (July Ocober 997), he Sepember 998 LCM debacle, he bursing of he high echnology Do-Com bubble of wih abou 30% losses of equiy values, evens such as 9/, sudden corporae collapses of he magniude of Enron and Lehman Brohers, and mos recenly, he 2007/08 credi crisis which has been considered o be he greaes since he Grea Depression. here has been a radical shif in he view held by policy makers, finance academics and praciioners who now feel ha exreme evens in financial markes canno be ignored as ouliers wih negligible probabiliy. In mainsream financial heory, exreme evens which occur wih small probabiliies have no been a maer of concern as in he dominan model of lognormal asse prices he probabiliy of exreme evens such as he sock marke crash of Ocober 987 is virually non-exisen. here has been a growing pragmaic and heoreical shif in ineres from he modelling of normal asse marke condiions o he shape and faness of he ails of he disribuions of asse reurns which characerize saisical models for exreme evens. Exreme value heory is a robus framework o analyse he ail behaviour of disribuions. Exreme value heory has been applied exensively in hydrology, climaology and also in he insurance indusry. Embrechs e. al. [997] is a comprehensive source on exreme value heory and applicaions. Despie early work by Mandelbro [963] on he possibiliy of fa ails in financial daa and evidence on he inapplicabiliy of he assumpion of log normaliy in opion pricing, a sysemaic sudy of exreme value heory for financial modelling and risk managemen has only begun recenly. 2 he objecive of his aricle is o use he Generalized Exreme Value (GEV) disribuion in he conex of European opion pricing wih he view o overcoming he problems associaed wih exising opion pricing models. Wihin he arrison and Pliska [98] asse pricing framework, he risk neural probabiliy densiy (RND) funcion exiss under an assumpion of no arbirage. By definiion of a no arbirage equilibrium, he curren price of an asse is he presen discouned value of is expeced fuure payoff given a risk-free ineres rae where he expecaion is evaluaed by he RND funcion. Breeden and Lizenberger [978] were firs o show how he RND funcion can be exraced from raded opion prices. he Black-Scholes [973] and lognormal based RND models have well known drawbacks. Firs, he implied volailiy smiles or smirks are inconsisen wih he consancy required in he lognormal case for volailiy across differen srikes for opions wih he same mauriy dae. Furher, his class of models canno explicily accoun for he negaive skewness and he excess kurosis of asse reurns. Since, Jackwerh and Rubinsein [996] demonsraed he disconinuiy in he implied skewness and kurosis across he divide of he 987 sock marke crash - a large lieraure has developed which aims o exrac he RND funcion from raded opion prices so ha he skewness and fa ail properies of exreme marke evens are beer capured han is he case in lognormal models. Pricing biases caused by lef skewness of asse reurns ha canno be capured in he implied lognormal asse pricing models are now well undersood (see,corrado and Su [996,997], Savickas As noed by Jackwerh and Rubinsein [996] in a lognormal model of asses prices, he marke crash on 9 Ocober 987 wih a 29% fall of S&P 500 fuures prices has a probabiliy of 0-60, an even which is unlikely o happen even in he life ime of he universe. 2 Embrechs e. al. [999], Mc Neil [999] and Embrechs [2000] consider he poenial and limiaions of exreme value heory for risk managemen. Dowd [2002] gives a good accoun of hese developmens and a recen survey of exreme value heory for finance can be found in Rocco [200]

3 [2002]). ypically, in periods when he lef skewness of asse prices increases, he Black-Scholes model will overprice ou-of-he-money call opions and underprice in-he-money call opions relaive o when here is greaer symmery in he disribuion funcion. his aricle shows how he opion price is highly sensiive o changes in he ail shape, which is disinc o is sensiiviy o he variance of he reurns disribuion. We find ha he raded opion price implied GEV model for he RND yields resuls ha srongly challenge radiionally held views on ail behaviour of asse reurns based on Gaussian disribuions which predicae simulaneous exisence of hin ails in boh direcions during all marke condiions. he GEV disribuion, which is governed by he ail shape parameer, is found o swich ail shape wih underlying marke condiions. During exreme marke drawdowns, a posiive value for he ail shape parameer resuls in significan skewness in he probabiliy mass of he GEV densiy funcion for losses and implies exreme price drops wih he large probabiliy mass on he righ and a runcaed ail in he oher direcion, implying an upper bound on possible gains. o dae, proposed opion pricing models inended o deal wih boh he fa ail and he skew in asse reurns have failed o highligh he above characerisic feaures of fa ailed disribuions. hey have also run ino problems ranging from a lack of closed form soluion, a large number of parameers needed or he lack of easy inerpreaion of implied parameers. hese facors have prevened many of hese models from being of pracical use in pricing and hedging opions or in risk managemen for exreme marke condiions. his aricle argues for he use of he Generalized Exreme Value (GEV) disribuion for asse reurns in an opion pricing model for he following reasons: (i) I can provide a closed form soluion for he European opion price. (ii) I yields a parsimonious European opion pricing model, wih only wo parameers o esimae, he ail shape parameer and he scale parameer. (iii) I provides a flexible framework ha subsumes as special cases a number of classes of disribuions ha have been assumed o dae in more resricive seings. he GEV disribuion encompasses he hree main classes of ail behaviour associaed wih he Fréche ype fa ailed disribuions and he hin and shor ailed Weibull and Gumbel classes. (iv) When he GEV disribuion is of Fréche ype, i exhibis a fa ail on he righ and a runcaed ail on he lef. Since exreme economic losses are more probable han exreme economic gains, we adop he Fréche disribuion o model exreme losses. o his end, we follow he pracice of he insurance indusry, Dowd [2002, p 272], and model reurns as negaive reurns. As a resul, when exreme evens are prominen, he GEV model yields a Fréche ype implied densiy funcion for negaive reurns, signifying higher probabiliies of price drops. (v) Mos significanly, he GEV opion pricing model can deliver he marke implied ail index for asse reurns. I is imporan o capure marke percepion of fa ailed behaviour in asse reurns in a manner which is inerspersed wih hin and shor ailed Gumbel and Weibull values for he ail index which characerize more normal marke condiions. ence, he marke implied ail index is found o be ime varying in a way ha mirrors he lack of invariance in he recursively esimaed - 3 -

4 ail index of asse reurns (see, Quinos, Fan and Phillips [200]) wih jumps in he fa ailedness in crisis periods. (vi) We show how he GEV opion pricing model removes he well known pricing biases associaed wih he Black-Scholes, by capuring he ime varying levels of skewness and kurosis. We also show how he GEV model yields superior pricing accuracy ou of sample, as GEV implied RNDs are more capable of capuring exreme marke condiions han oher opion pricing models. (vii) aving obained a closed form soluion for he opion pricing model, we can also obain a closed form soluion for he new greek in he lexicon of opion pricing, which measures he sensiiviy of he opion price o he ail index. (viii) he closed form dela hedging formulaion can also be given. his aricle covers he firs six feaures lised above of he GEV RND model of opion pricing and we leave he las wo for furher work. We will now briefly commen on how he GEV RND based opion pricing model fis ino he large edifice, given in Exhibi below, buil from he differen mehods used for he exracion of he implied disribuions and heir respecive opion pricing models ha have arisen since he work of Breeden and Lizenberger [978]. Based on Jackwerh [999] survey, he differen mehods can be classified ino hree main caegories: parameric, semi parameric and non-parameric. Parameric mehods can be divided ino hree sub-caegories: generalized disribuion mehods, specific disribuions and mixure mehods. Generalized disribuion mehods inroduce more flexible disribuions wih addiional parameers beyond he wo parameers of he normal or lognormal disribuions. Wihin his subcaegory, Aparicio and odges [998] use generalized Bea funcions of he second kind, which are described by four parameers, and Corrado [200] uses he generalized Lambda disribuion. Under he specific disribuions being assumed for he RND funcion, he Weibull disribuion is used by Savickas [2002], and he skewed Suden- by de Jong and uisman [2000]. he Variance Gamma disribuion used by Madan, Carr and Chang [998], and Levy processes used among ohers by Maache, Nische and Schwab [2004] are more recen specificaions wih hese mehods having parameers ha can conrol fa ails and skewness of he asse price. Up o seven parameers are associaed wih hese models. Finally, he hird sub-caegory wihin parameric mehods is he mixure mehods, which achieve greaer flexibiliy by aking a weighed sum of simple disribuions. he mos popular mehod here is mixure of lognormals. Richey [990] and Gemmill and Saflekos [2000] use wo lognormals, and Melick and homas [997] use hree lognormals. One problem associaed wih he mixure of disribuions is ha he number of parameers is usually large, and hus hey may overfi he daa. For example, he mixure of wo lognormals needs o esimae five parameers. Under he caegory of semi parameric mehods, he ypergeomeric funcion was used by Abadir and Rockinger [997], and expansion mehods such as he Gram-Charlier and Edgeworh expansions, respecively, were used by Corrado and Su [996] and Corrado and Su [997]. he nonparameric mehods can be divided again ino hree groups: kernel mehods, maximum-enropy mehods, and curve fiing mehods. Kernel mehods, implemened in Ai-Sahalia and Lo [998], are - 4 -

5 relaed o regressions since hey ry o fi a funcion o observed daa, wihou specifying a parameric form. Second, he mehods based on maximum-enropy used by Buchen and Kelly [996] find a nonparameric probabiliy disribuion ha ries o mach he informaion conen, while a he same ime saisfying cerain consrains, such as pricing observed opions correcly. In he hird group in his caegory, here are he curve fiing mehods ha ry o fi he implied volailiies wih some flexible funcion. he mos popular of hese is Shimko [993] who inroduced he concep of smoohed implied volailiy smiles which involved fiing ypically a cubic or low order polynomial spline o obain he middle porion of he RND funcion. he ails of he RND funcion were modelled as log normal. his approach was improved by Bliss and Panigirzoglou [2002] wih he use of a smoohing spline whils reaining log normal ails. Figlewski [200] made an advance on his by appending ails from he GEV disribuion which are able o reflec exreme marke condiions. Exhibi Classificaion of mos common RND esimaion mehods Generalized Bea funcions (Aparicio and odges [998]) Generalized disribuions Generalized Lambda Disribuion (Corrado [200]) Generalized Exreme Value (GEV) disribuion Weibull disribuion (Savickas [2002]) Parameric mehods Specific disribuions Skewed Suden- (de Jong and uisman [2000]) Variance Gamma (Madan e al [998]) Lévy process (Maache e al [2004]) Mixure mehods Mixure of wo lognormals (Richey [990]) Mixure of hree lognormals (Melick and homas [997]) ypergeomeric funcions (Abadir and Rockinger [997]) Semi parameric mehods Gram-Charlier expansions (Corrado and Su [996]) Edgeworh expansions (Corrado and Su [997]) Kernel mehods (Aï-Sahalia and Lo [998]) Non-parameric mehods Maximum enropy mehods (Buchen and Kelly [996]) Curve fiing mehods (Shimko [993], Figlewski [200]) - 5 -

6 he model presened in his aricle, as highlighed in Exhibi, falls in he general caegory of parameric models, and more specifically, wihin he sub-caegory of generalized disribuions. In order o esimae ail behaviour a high confidence levels, such as 99%, many non-parameric mehods for RND esimaion fail o capure ail behaviour of he disribuions because of sparse daa for opions raded a very high or very low srikes prices. ence, parameric models have become unavoidable. his, however, replaces sampling error wih model error. In he nex secion, we give a brief inroducion o Exreme Value heory and presen he Generalized Exreme Value (GEV) disribuion and is properies o indicae how he flexibiliy of his hree parameer class of disribuions can capure skew and fa ails as and when dicaed by he daa wih no a priori resricions on he class of disribuion. his daa driven selecion of he ail index miigaes model error. he res of he aricle is organized as follows. We develop he GEV opion pricing model and he closed form soluions for he arbirage free European call and pu opion prices are derived for he GEV based RND funcion. We hen proceed o discuss he componens of he closed form soluion and heir heoreical properies in erms of moneyness and hen ail shape parameer. he empirical secion repors on he resuls for he esimaed implied GEV RND funcion and for is parameers based on he FSE 00 European opion price daa from 997 o he in sample fi of he posulaed GEV opion pricing model is compared wih he benchmark Black-Scholes one and is found o be superior a all levels of moneyness and a all ime horizons, removing he well known price bias of he Black- Scholes model. Ou of sample pricing ess show ha he GEV provides superior pricing performance compared o Black-Scholes, for one day ahead forecass. he analysis of he ime series characerisics of he implied ail index is given and he role of implied RND funcions in even sudies surrounding periods of exreme price falls of he FSE-00 index is also discussed. Finally, we make concluding remarks and discuss fuure work. EXREME VALUE EORY AND E GEV DISRIBUION Unlike he normal disribuion ha arises from he use of he cenral limi heorem on sample averages, he exreme value disribuion arises from he limi heorem of Fisher and ippe [928] on exreme values or maxima in sample daa. he class of GEV disribuions is very flexible wih he ail shape parameer (and hence he ail index defined as α= - ) conrolling he shape and size of he ails of he hree differen families of disribuions subsumed under i. hese hree families of disribuions can be nesed ino a single parameric represenaion, as shown by Jenkinson [955] and von Mises [936]. his represenaion is known as he Generalized Exreme Value (GEV) disribuion and is given by: ( ( + ) ) F ( x) = exp x wih + x > 0, 0 (.a) Applying he formula ha 0 x ( + x) e, as 0 we have: x F ( x) = exp( e ) (.b) - 6 -

7 he sandardized GEV disribuion, in he form in von Mises [936] (see, Reiss and homas [200], p. 6-7), incorporaes a locaion parameer µ and a scale parameer, in addiion o he ail shape parameer,, and is given by: and ( x µ ) F, µ, ( x) = exp + wih ( x µ ) ( µ ) + x > 0 0 (2.a) F ( x) = exp( e ) wih = 0 (2.b) 0, µ, he corresponding probabiliy densiy funcions obained by aking he derivaive of he disribuion funcions, are respecively: and ( x µ ) ( x µ ) f, µ, ( x) = + exp + 0 ( x µ ) / ( x µ ) / f 0, µ, ( x) = e exp( e ) = 0 (3.a) (3.b) We will now discuss how he ail shape parameer,, deermines boh he higher momens of he densiy funcion and also he skew in he probabiliy mass leading o runcaion poins in he disribuion. he ail shape parameer =0 yields hin ailed disribuions wih he ail index α= - being equal o infiniy, implying ha all momens of he disribuion are eiher finie or zero. 3 When = 0, he GEV disribuion belongs o he Gumbel class and includes he normal, exponenial, gamma and lognormal disribuions, where only he lognormal disribuion has a moderaely heavy ail. he Gumbel class has zero skew in he probabiliy mass and displays symmery in he righ and lef ails. Furher, as seen in equaion (2.b) here are no condiions runcaing he disribuion in eiher direcion for values of x. he disribuions associaed wih > 0 are called Fréche and hese include well known fa ailed disribuions such as he Pareo, Cauchy and Suden- disribuions. Finally, in he case where < 0, he disribuion class is Weibull. 4 hese are shor ailed disribuions wih finie upper bounds and include disribuions such as uniform and bea disribuions. In disribuions for which 0, he equaliy condiion in equaion (2.a) imposes a runcaion of he probabiliy mass and a disinc asymmery in he righ and lef ails such ha when he probabiliy mass is high a one ail signifying non-negligible probabiliy of an exreme even in ha direcion, here is an absolue maxima (or minima) in he oher direcion beyond which values of x have zero probabiliy. As shown in Reiss and homas [200], kurosis of he Fréche disribuion becomes infinie a 0.25 (he ail index, α 4), and all higher momens including kurosis and he righ skew become infinie a 0.33 (he ail index, α 3). Even for small posiive values of, approximaely a abou = 0., he rae of growh of 3 he general rule is ha he n h and higher momens fail o be finiely inegrable if he ail index is smaller han n. When 0, all momens are finie or zero. owever, when 0, only momens up o he ineger par of he ail index, α =/, exis wih all oher momens being infinie. 4 ere we make reference o he Weibull disribuion as defined in he conex of exreme value heory, Embrechs [997, p54]

8 skewness and kurosis of he disribuion, wih boh fas approaching infinie growh, resuls in a concenraion of he probabiliy densiy of he Fréche disribuion a he righ ail. hus, as increases wih > 0, he runcaion poins a he lef ail a which here is zero probabiliy become more sringen. Noe for < 0, for he Weibull class of disribuions, here is increased probabiliy mass on he lef ail and a runcaion poin given by he inequaliy in equaion (2.a) a he righ ail. owever, i is well known (see, Reiss and homas [200]) ha a abou = -0.3, he Weibull disribuion has zero skew and is indisinguishable from a Gumbel disribuion. As he probabiliy of exreme economic losses are more likely han exreme gains, economic losses are modelled as a Fréche disribuion wih high probabiliy mass on he righ ail. Exhibi 2a below illusraes he GEV densiy funcions for negaive asse reurns for each of he hree classes of disribuions ha he GEV can ake based on he shape parameer. Noe, ha he hree graphs only differ in he value of (he values considered for are 0.3, 0, -0.3), having he same value for locaion (µ=0) and scale (=0.2) parameers. he iniial sock price is assumed o be 00. he corresponding densiy funcions for he price in each of he hree cases for he ail shape parameer are shown in Exhibi 3. Noe, he lef skew in he price densiy funcion is greaes in Exhibi 3c, for he case when > 0, and he negaive reurns densiy funcion belongs o he GEV-Fréche class. Given he value being assumed, = 0.3, in Exhibis 2c and 3c, as noed above, here is infinie kurosis and a very sringen runcaion on posiive reurns exceeding or for he prices o rise above Likewise, for = in (2.a,3.a), we have zero probabiliy for negaive reurns o be greaer han and he price o fall below hese upper and lower bounds on reurns and prices implied by he GEV disribuion play an imporan role in he analysis ha follows. As will be shown laer, for he range of values for he implied ail shape parameers for 30 day reurns on he FSE-00 index ha we exrac on a daily basis from opion prices over he sample period from , he maximum value we obain for is As his implies a ail index value of α = 8.33, i is clear ha his guaranees finie skewness and kurosis for he risk neural densiy funcion for he enire sample period. Furher, on using equaion (2.a), precise runcaions values under he RND Q- measure can be deermined for he levels of he sock index and for he reurns on i. In he conex of he opion pricing model i is imporan o verify ha he runcaed values implied by a GEV based RND, ie. Q-impossible evens are also no P-possible in erms of he empirically realized values for prices and reurns. 6 his will be invesigaed in he nex secion. Exhibi 2 Densiy funcions for negaive reurns 5 By rearranging he inequaliy in equaion (2.a) and using he values being assumed for he GEV parameers, he runcaion values denoed by x* for negaive reurns in Exhibis (2.c, 3.c)) and (2.a,3.a) are deermined from x*> µ /. 6 We are graeful o Sephen Figlewski for bringing his o our aenion

9 (a) = (b) = 0 (c) = Noes: Densiy funcion of negaive reurns as modeled by he (a)gev-weibull, (b)gev-gumbel and (c)gev-fréche. Exhibi 3 Densiy funcions for prices (a) = (b) = 0 (c) = Noes: Corresponding densiy funcion for prices where negaive reurns have been modeled as (a) GEV- Weibull, (b) GEV- Gumbel and (c) GEV- Fréche. E GEV OPION PRICING MODEL Arbirage Free Opion Pricing and he Risk Neural Densiy Le S denoe he underlying asse price a ime. he European call opion wih price C is wrien on his asse wih srike K and mauriy. We assume he ineres rae r is consan. Following he arrison and Pliska [98] resul on he arbirage free European call opion price, here exiss a risk neural densiy (RND) funcion, g(s ), such ha he equilibrium call opion price can be wrien as: r( ) Q r ( ) ( K ) = e E [ max( S K,0) ] = e ( S K ) K C g( S ) ds (4) Q ere, Q is he risk neural measure and [ ] E is he risk-neural expecaion operaor condiional on informaion available a ime, g(s ) is he risk-neural densiy funcion of he underlying a mauriy. Similarly, he arbirage free opion pricing equaion for a pu opion is given by: K r( ) [ max( K S,0)] = e ( K r ( ) Q P ( K) = e E S ) g( S ) ds (5) 0 In an arbirage-free economy, he following maringale condiion mus also be saisfied: S ( S ) r( ) Q = e E (6) European Call and Pu Opion Price wih GEV reurns - 9 -

10 We assume ha he RND funcion g(s ) in (4) for a holding period equal o ime o mauriy of he opion is represened by he GEV disribuion. We derive closed form soluions for he call and pu opion pricing equaions by analyically solving he inegrals in (4) and (5). For he purpose of obaining an analyic closed form soluion, i was found necessary o define reurns as simple reurns. 7 We define simple negaive reurns as follows: L S S S = R = = (7) S S In keeping wih he exreme value disribuion modelling of economic losses, L is assumed o follow he GEV disribuion given in (3.a), 0 given by: ) = +,and hence he densiy funcion for he negaive reurns is ( L µ ) ( L µ ) exp + ( L f (8) Noe, he relaionship beween he densiy funcion for L in (8) and he RND funcion g(s ) in (4) for he underlying price S is given by he general formula: g ( S ) f ( L ) = f ( L ) L S S = (9) On subsiuing (8) ino (9), we obain he RND funcion of he underlying price in erms of he GEV densiy funcion as in equaion (3.a) : ) = + S ( L µ ) ( L µ ) exp + ( S g (0) wih S + µ > S ( µ ) = + 0 L () We will firs consider he case when >0 and 0 < <. 8 As already discussed, in his case he negaive reurns disribuion is Fréche and his implies ha he price RND funcion g(s ) in (0), in order o saisfy he condiion in (), is runcaed on he righ. ence, he upper limi of inegraion for 7 Noe, simple reurns can give rise o he heoreical possibiliy of negaive sock prices when > 0. owever, for purposes of opion pricing his does no pose a problem as for he call price he lower limi of inegraion K for he sock price in equaion (4) is always posiive and likewise for he pu price he lower limi of inegraion for he sock price in equaion (5) is zero. Addiionally, numerical resuls (no repored in his aricle) show ha he implied GEV parameers (, ) obained when using simple reurns are no saisically differen o he ones obained using log reurns. 8 he condiion 0< < is necessary o rule ou he case ha he firs momen for he sock price a mauriy is infinie and he opion value becomes infinie. In his aricle all cases of >0 will be consrained in his way. he closed form soluion for he call opion for he case when 0< < is idenical o he one obained for he case when < 0. his is also rue for he closed form soluion for pu opion prices. Appendix B derives he closed form soluions for he call and he pu opions in he case of =

11 he call opion price in (4) becomes S ( µ + ) equaion in (4), we have:. 9 Subsiuing g(s ) in (0) ino he call price ( L µ ) ( L µ ) S ( µ + ) r( ) ( ) ( ) C = + K e S K exp + ds (2) K S Consider he change of variable: S ( ) y = + L = + µ µ S (3) Under his change of variable, he underlying price S and ds can be wrien in erms of y as follows: S = S µ ( y ) and ds S dy = (4) Also, he densiy funcion in (0) for he underlying price a mauriy in erms of y becomes: / ( y ) exp( ) g( y) = y S (5) Noe ha under he change of variable he lower limi of inegraion for he call opion equaion in (2) becomes: K = + µ S (6) he upper limi of inegraion in (2) becomes 0. Subsiuing for S and ds as defined in (4) ino (2), and using he new limis of inegraion we have: C e r ( ) = S 0 µ S ( y ) K ( y ) ( exp y ) S dy (7) Simplifying and rearranging (7) we have: 9 On he oher hand, when < 0 he GEV densiy funcion for S is runcaed on he lef, and herefore, he lower limi of inegraion for he call opion price in (4) becomes max[k, S ( - µ + /)] and he upper limi remains. owever, he closed form soluions for he call opion are idenical for boh cases when > 0 and < 0. his also holds for pu opion prices. - -

12 C e r ( ) = 0 S = S µ 0 y ( y ) K ( y ) exp( y ) 0 ( y ) exp( y ) dy S µ + K ( y ) exp( y ) S = ψ S µ + K ψ 2 / he inegral ψ in (8) above can be solved by applying he change of variable = y, and hen i can be evaluaed in erms of he incomplee Gamma funcion, yielding he following soluion: ψ 0 = y exp( y ) dy = Γ( ), dy (8) (9) dy he soluion of inegral ψ 2 in (8) is: 0 0 = ( y ) exp( y ) dy = [ exp( y )] = ( ( ) ψ 2 exp (20) Combining resuls for ψ andψ 2, we obain a closed form for he GEV call opion price: / r( ) S / C = Γ( ) ( K) e, S µ + K ( e ) (2) Grouping he erms wih S ogeher we have: r( ) C ( K) = e S K e (22) ( µ + ) e Γ(, ) heoreically, he applicaion of he Girsanov heorem (see, Nefci [2000]) o opion pricing implies ha he empirical disribuion and he risk neural disribuion need o have he same suppor. By he Girsanov heorem, he price levels and he size of reurns ha are Q-impossible due o he applicaion of he runcaion condiion in () should no be P-possible, and vice versa, in erms of he realized hisorical prices and reurns. We find ha he condiions of he Girsanov heorem are saisfied for he sample period for which he implied GEV based RND is exraced from opion prices. he analysis of his is given in Appendix D. Following similar seps, we can also derive a closed form soluion for he pu opion price under GEV reurns. 0 Deails of his derivaion can be found in he Appendix A, which yields he following equaion: 0 Once he call pricing formula is derived, one could simply obain he pu pricing formula using he pu-call pariy relaionship. We numerically verified ha he independenly derived call and pu pricing formulas saisfy pu-call pariy

13 h h ( e ) S ( µ + )( e e ) Γ(, h, ) r( ) P( K) = e K e (23) = µ where + ( ) > 0 h. Noe ha h is a consan, given a se of parameers µ,, and. In he following secions, we will analyse he properies of he GEV RND based closed form soluions for he call and pu opions under differen moneyness condiions and values for he ail shape parameer. Analysis of he GEV call opion pricing model his secion aims o give some insighs ino he closed form soluion for he GEV based call opion pricing equaion given in (22), which has wo componens and respecive probabiliy weighs involving S and K. hese wo componens can be inerpreed along he same lines as he Black-Scholes model. he key o undersanding he GEV opion pricing formula lies wih he erm, + K S µ e = e (24) his erm is he cumulaive GEV disribuion funcion as defined in (2.a) for he sandardized moneyness of he opion defined as (S K)/S. ence, i corresponds o he risk neural probabiliy p of he call opion being in he money a mauriy. For a given se of implied GEV parameers {µ,, } we can work ou (see, Exhibi 4) he range of exercise prices K in relaion o he given S which yield: e = for deep in-he-money call opions, e = 0 for deep ou-of-he-money call opions, and 0 < e < for all oher cases. Exhibi 4 below plos he probabiliy p= e of exercising he opion a mauriy, given by he GEV model wih wo differen values of, and also for he Black-Scholes model. 2 When >0, he densiy funcion of losses is Fréche, and hus, he implied price densiy funcion is lef skewed wih a fa ail on he lef, as shown respecively in Exhibis 2c and 3c. Since he laer implies here is a higher probabiliy of downward moves of he underlying han in he Black-Scholes case, we see from Exhibi 4 how he probabiliy of exercising he call opion when > 0 approaches much slower han for he Black-Scholes model. Exhibi 4 Probabiliy of he call opion being in he money a mauriy Recall ha in he case of he Black-Scholes model he probabiliy of he opion being in he money a mauriy is given by N(d 2), 2 where N() is he sandard cumulaive normal disribuion funcion, and d 2 = [ln( S / K ) + ( r / 2) ]/ 2 o make he hree cases comparable, we use he same raded call opion price daa o esimae he GEV model and he Black- Scholes model. hen, o obain he second case for he GEV model, we fix o be equal o he iniial esimae, bu wih opposie sign, and esimae he oher wo GEV parameers

14 Noes: he probabiliy of he call opion being in he money a mauriy is given by p GEV =exp(- -/ ) for he GEV case, and by p B-S =N(d 2 ) for he Black-Scholes. he posiive and negaive values of used for he GEV disribuion are 0.6 and -0.6 respecively. On he oher hand, when < 0, he GEV densiy of he losses is of Weibull ype, and hus he implied price densiy funcion is righ-skewed, resuling in a higher probabiliy of upward moves. herefore, he probabiliy p of exercising he opion as we lower he srike price K reaches faser han in he Black-Scholes case. Noe ha for high srike prices and for any value of 0, he probabiliy of he opion being in he money goes o zero faser han for he Black-Scholes case. When he call opion is deep in-he-money (IM) wih K << S and e =, he call price converges o a linear funcion of he expeced payoff (see Appendix C for proof). hus, Q r ( ) ( E [ S ] K ) = S e K r( ) C ( K) = e (25) Q ere [ S ] GEV GEV E is he condiional firs momen of he price RND funcion, which by he maringale condiion in (6) equals S. For his range of srike prices, he opion prices obained wih he GEV model converge o hose given by he Black-Scholes model. When he opion is deep ou of he money, hen K >> S and e = 0, and i is easy o verify ha he call price is zero. Exhibi 5 displays he call opion prices obained from he GEV model and he Black-Scholes model. he Black-Scholes model overprices he ou-of-he-money (OM) call opions relaive o he GEV model in boh cases. In he OM case, he GEV model yields higher values of call prices when < 0 han when > 0. his is because when < 0, upward movemens in he underlying price are more likely and he price densiy is runcaed on he lef (see Exhibi 3a). On he oher hand, when > 0 downward movemens in he price are more likely and he price densiy funcion is runcaed on he righ (see Exhibi 3c). ence, in he OM region of high exercise prices, K, he GEV price wih > 0 gives he lowes prices for he call opion

15 Exhibi 5 Call opion prices for he GEV model and he Black-Scholes model Noes: he posiive and negaive values of used for he GEV disribuion are 0.6 and -0.6 respecively. For in-he-money (IM) opions, as seen in Exhibi 5, he Black-Scholes model under prices call opions when compared o he GEV model, and he GEV model gives higher opion prices when > 0 han when < 0. his can be explained in erms of he asymmery in he peakedness of he wo densiies. When > 0, he RND funcion for he price is lef skewed, wih peakedness a higher values of he underlying han when < 0. For a-he-money (AM) opions, he prices given by boh models are approximaely he same. Noe ha for deep IM opions, i.e. for much lower values of K (no shown in he graph) boh GEV and Black-Scholes prices converge o he presen discouned value of he inrinsic value of he opion, increasing linearly as K falls. Analysis of he GEV pu opion pricing model he analysis for he closed form soluion of he GEV pu opion pricing model in equaion (23) is analogous o wha was done in he case of he call opion. he probabiliy of a pu being in he money is given by h e e e (26) ere, noe h e is approximaely equal o and hence (26) is one minus he probabiliy of he call being in he money a mauriy. In Exhibi 6, while considering he case of a Fréche disribuion for losses wih > 0, for low srike prices relaive o he underlying, we have a greaer probabiliy of he pu opion being in he money a mauriy as compared o eiher he Black-Scholes case or he GEV case when < 0. Exhibi 6 Probabiliy of he pu opion being in he money a mauriy - 5 -

16 Noes: he probabiliy of being in he money for he pu opion a mauriy is given by p GEV = - exp(- - / ) for he GEV case, and by p B-S =N(-d 2 ) for he Black-Scholes. he posiive and negaive values of used for he GEV disribuion are 0.6 and -0.6 respecively. Exhibi 7 below displays he pu opion prices obained wih he GEV model along wih he Black-Scholes model. he Black-Scholes model subsanially underprices he ou-of-he-money (OM) pu opions relaive o he GEV model. he GEV model yields higher values of OM pu prices when > 0 han when < 0. For in-he-money (IM) pu opions, he Black-Scholes model only marginally overprices pu opions wih respec o he GEV model. he GEV model gives higher prices for IM pu opions when < 0 han when > 0. For a-he-money (AM) opions, he prices given by boh he GEV and he Black-Scholes models are approximaely he same. Noe ha for deep IM pu opions, boh GEV and Black-Scholes prices converge o he presen discouned value of he inrinsic value of he opion, e r ( ) K S, which increases linearly wih K. Exhibi 7 Pu opion prices for he GEV model and he Black-Scholes model Noes: he posiive and negaive values of used in he case of he GEV model are 0.6 and -0.6 respecively

17 RESULS Daa descripion he daa used in his sudy are he daily selemen prices of he FSE 00 index call and pu opions published by he London Inernaional Financial Fuures and Opions Exchange (LIFFE). hese selemen prices are based on quoes and ransacions during he day and are used o mark opions and fuures posiions o marke. Opions are lised a expiry daes for he neares four monhs and for he neares June and December. FSE 00 opions expire on he hird Friday of he expiry monh. he FSE 00 opion srikes are in inervals of 50 or 00 poins depending on ime-o-expiry, and he minimum ick size is 0.5. here are four FSE 00 fuures conracs a year, expiring on he hird Friday of March, June, Sepember and December. he LIFFE exchange quoes selemen prices for a wide range of opions, even hough some of hem may have no been raded on a given day. In his sudy we only consider prices of raded opions, ha is, opions ha have a non-zero raded volume on a given day. he daa was also filered o exclude days when he cross-secion of opions had less han hree opion srikes. Also, opions whose prices were quoed as zero, had less han one week o expiry, or more han 20 days o expiry were eliminaed. Finally, opion prices were checked for violaions of he monooniciy condiion. 3 he period of sudy is from 2-Jan-997 o -Jun Exhibi 8 below summarizes he average number of raded opion prices available on a daily basis, across boh srikes and mauriies, for each of he years in he period under sudy, including boh call and pu opions. We can see how he number of raded conracs has increased subsanially hrough ime, from an average of 45 daily raded opion prices in 997 o 59 such prices in he range of srikes wih opions raded has also widened hrough ime. Exhibi 8 Summary daa on FSE-00 Index opion prices Period Number of opion prices Minimum srike Maximum srike (daily average) Monooniciy requires ha he call (pu) prices are sricly decreasing (increasing) wih respec o he exercise price. A small number of opion prices ha did no saisfy his condiion were removed from he sample

18 All Years Noes: Average number of raded opion prices available per day, minimum and maximum srike price wih opions rade per year (Jan 997-June2009). he European-syle FSE00 opions, hough hey are opions on he FSE 00 index, can be considered as opions on FSE-00 index fuures, because he fuures conrac expires on he same dae as he opion. herefore, he fuures will have he same value as he index a mauriy, and can be used as a proxy of he underlying FSE 00 index. By using his mehod, we avoid having o use he dividend yield of he FSE 00 index, and he maringale condiion in (6) becomes: Q ( S ) F = E (27) ere F is he price of he FSE 00 fuures conrac a, and S is he FSE 00 index a mauriy. his maringale condiion can be used o reduce he number of parameers in he GEV model from 3 o 2. his is analogous o he procedure in he Black-Scholes model where he mean of he disribuion is obained from he maringale condiion and only he volailiy parameer needs o be esimaed. In he GEV case, he mean of he disribuion does no direcly correspond o he locaion parameer. Insead, he mean of he GEV disribuion, as shown in Reiss and homas [200], is a funcion of all hree parameers and is defined as µ + ( Γ( ) )/ ). We can use his definiion of he GEV mean ogeher wih (6) o express he locaion parameer µ in erms of he fuures price curren spo price, and he GEV scale and ail shape parameers and : F,, ( ) F, Γ µ = S (28) he risk-free raes used are he Briish Bankers Associaion s a.m. fixings of he 3-monh Shor Serling London InerBank Offer Rae (LIBOR) obained from he websie Even hough he 3-monh LIBOR marke does no provide a mauriy-mached ineres rae, i has he advanages of liquidiy and of approximaing he acual marke borrowing and lending raes faced by opion marke paricipans (Bliss and Panigirzoglou [2004]). he opion daa used in his sudy can be divided ino 6 moneyness caegories given in Figlewski [2002]. 4 Exhibi 9 below repors he number of observaions for call and pu opions in each caegory of moneyness and mauriy. Exhibi 9 Number of observaions in each mauriy and moneyness caegory 4 Figlewski [2002] explains ha a measure of opion moneyness should include an adjusmen for volailiy and mauriy. Following his definiion, we calculae he moneyness of an opion as r BS BS ln( S / Ke ) /( ), where is he Black- Scholes implied volailiy for each opion

19 Number of Observaions Subsample Calls Pus Mauriy < 30 days 25,70 3, o 60 days 24,553 3, o 90 days 5,463 9, o 20 days 8,59 9,905 Moneyness deep OM 8,42 7,895 OM 23,507 38,043 AM 30,086 26,808 IM 8,7 5,790 deep IM 3,70 4,72 oal 73,885 92,708 Noes: Number of observaions of raded opions for differen mauriy and moneyness caegories (January 997- June 2009). ere, moneyness for a given opion indicaes how many sandard deviaions,, he srike price is away from he curren underlying price in erms of he volailiy, mauriy of he opion. An opion is deep ou-of-he-money (Deep OM) if i is more han.5 ou of he money, and similarly, i is deep in-he-money (Deep IM) if i is more han.5 in-he-money. An opion is classified as being a he money (AM) if i is 0.5 in eiher direcion of OM and IM. An addiional classificaion is done for mauriy, in erms of days o expiraion. Noe ha here are opions daa available for ime o expiraion longer han 20 days, bu he number of prices available for such long ime horizons is small and he opions are raded less frequenly. As can be seen in Exhibi 9, he shor o medium erm ime o mauriy, he firs wo groups, have he greaes number of daa poins, for boh pus and calls. In erms of moneyness, he larges number of observaions is found in he OM and AM caegory, while he deep IM has he leas number of observaions for boh pus and calls (IM opions are ypically very expensive, as he opion premium includes he inrinsic value, and hus are no raded ofen). Empirical Mehodology For each quarerly expiraion dae in our daa period, a oal of 49 from March 997 o March 2009, a arge observaion dae was deermined wih horizons of 90, 60, 30 and 0 days o mauriy. If no opions were raded on he arge observaion dae, he neares dae wih raded opions was used. All raded opion prices available for each arge observaion dae were used, subjec o he filers discussed above, across all srikes and across all mauriies, giving a one year consan horizon implied RND. 5 he implied RND was derived using he GEV and he Black-Scholes opion pricing models. 5 In order o esimae a single scale parameer o fi he prices of opions across muliple horizons, we need o annualise he scale parameer in he pricing equaions. his is similar o he procedure used in he Black-Scholes model such ha he implied volailiy parameer represens an annualised value. o achieve his, a he esimaion sage, we replace he GEV scale parameer sigma by, where is he ime o mauriy of he opion, in number of years

20 For each of hese arge observaion daes a single implied RND was fied using boh pu and call prices. he GEV model was esimaed by minimizing he sum of squared errors (SSE) beween he opion prices D ~ given by he analyical soluion of he GEV opion pricing equaions in (22) and (23) and he observed raded opion prices D (including boh calls and pus) wih srikes K i, as indicaed below: N SSE( ) = min ζ, i= ~ 2 ( D ( K ) D ( K )) i i (29) Noe ha for he GEV model, we minimise he sum of squared errors wih respec o only wo GEV parameers, ie. scale and ail shape parameers, and. We use equaion (28) o subsiue ou he locaion parameer µ which has been derived as funcion of he fuures price, F,, curren spo price, and hese wo parameers. For he Black-Scholes model, we likewise derive a single implied volailiy parameer using boh call and pu prices. he opimizaion was performed using he nonlinear leas squares algorihm from he Opimizaion oolbox in MaLab. In sample pricing performance he in sample pricing performance ess consis of esimaing he implied densiies a ime, by using opion prices a ime as well, and hen analysing how well he model fis he same opion prices. he pricing performance is repored in erms of he roo mean square error RMSE, which represens he average pricing error in pence per opion: SSE( ) RMSE( ) = (30) N For each mauriy horizon (ie. 0, 30, 60, 90 days) he average pricing error is aken over a oal of 49 quarerly arge observaion daes from March 997 o March he analysis ha follows in Exhibi 0 repors he average pricing errors in erms of RMSE for each of hese horizons used, o highligh some of he ineresing pricing biases ha are observed. he GEV opion pricing model ouperforms he Black-Scholes model for all ime horizons, and for boh pus and calls. In paricular, he GEV model removes he large pricing bias ha he Black- Scholes model exhibis for opions far from mauriy. For a 90 day horizon, he Black-Scholes model has an average pricing error of 20.7 pence, while he error for he GEV is almos hree imes smaller, a 7.44 pence. Boh models display an improvemen in performance as ime o mauriy decreases. For close o mauriy opions, a a 0 day horizon, he GEV model coninues o produce lower pricing errors, alhough he difference beween he wo models becomes smaller. Anoher observaion is ha he Black-Scholes model exhibis larger pricing errors for pus han for calls. In conras, he GEV model exhibis similar sized errors for pus and call conracs. As anicipaed from discussion on he

21 GEV pu opion resul, i is imporan o noe ha for pu opions, he Black-Scholes model suffers a far greaer deerioraion in pricing performance when compared o he GEV model. While on average, across all mauriy days, he difference he errors for calls and pus for he GEV model is only 0.2 pence, for he Black-Scholes model, ha difference is subsanially larger a 2.36 pence, mosly driven by he differences beween pu and call errors for far from mauriy opions. Exhibi 0 In-sample pricing performance 90 days 60 days 30 days 0 days All days BS GEV BS GEV BS GEV BS GEV BS GEV Calls Pus All Noes: In-sample pricing performance of he Black-Scholes (BS) and he GEV models, in erms of Average Roo Mean Square Error for opion prices in pence, for opions wih horizons of 90, 60, 30 and 0 days o mauriy. Analysis of he in-sample pricing bias I has been well documened ha he Black-Scholes model exhibis a pricing bias for ou of he money and in he money opions, while pricing more accuraely a he money opions (Rubinsein, 985). he pricing bias is defined in equaion (3) as he deviaion of he model esimaed price wih respec o he observed marke price for each opion conrac: Price bias = Marke price Esimaed price (3) ere we ake he individual opion pricing errors obained from he esimaion done in he empirical mehodology secion, and repor he average price bias across moneyness levels using a spline mehod. 6 he average pricing bias for call opions is ploed below in Exhibi for 90 and 0 day ime horizon. For he 90 day ime horizon, and in keeping wih he resuls obained in he previous secion, he Black-Scholes model shows more deerioraion in pricing accuracy for far from mauriy conracs han for close o mauriy ones. A far from mauriy, he Black-Scholes model underprices IM call opions (moneyness from +0.5 o +.5) by over 5 pence, while i overprices OM call opions (moneyness from -0.5 o -.5) by around 20 pence. On he oher hand, he price bias for he GEV model appears o be much less dependen on he moneyness levels, delivering much lower price bias across all moneyness levels. For he 0 day ime horizon, we can see ha for close o mauriy call opions, he Black-Scholes model exhibis he same paern of price bias as for far from mauriy opions, bu he magniude of hese price biases is much smaller. In line wih resuls in he previous secion, boh models display a reducion in pricing bias as ime o mauriy decreases, and exhibi similar pricing biases oscillaing beween around +6 pence and 6 pence. 6 Given ha a differen arge observaion daes, we have differen moneyness levels, we fi a spline o he pricing error observaions as a funcion of moneyness on each day, and ake he average of hese splines across he 49 arge observaion daes for each horizon. Noe we do no show price biases ouside he [-2,+2] moneyness range as here are usually oo few daa poins o obain meaningful averages, bu model prices end o converge o marke prices in he limis, eiher collapsing o 0 for very deep OM or equalling he inrinsic value for very deep IM

22 Exhibi Average call price bias in erms of moneyness 35 a) 90 days o expiraion calls Pricing bias 25 BS GEV Moneyness 35 b) 0 days o expiraion calls Pricing bias 25 BS GEV Moneyness Exhibi 2 below display he pricing bias for pu opions. For far from mauriy pu opions, a 90 days o mauriy, he Black-Scholes model overprices IM pu opions (moneyness from -0.5 o -.5) by over 0 pence, while underprices OM pu opions (moneyness from +0.5 o +.5) by up o 30 pence. On he oher hand, he GEV model exhibis a small pricing bias across he board. For close o mauriy opions, he char for a 0 day ime horizon shows how boh models exhibi price biases of similar magniude of around ±6 pence. Exhibi 2 Average price bias for pus in erms of moneyness

23 35 a) 90 days o expiraion pus Pricing bias 25 BS GEV Moneyness 35 b) 0 days o expiraion pus Pricing bias 25 BS GEV Moneyness OU OF SAMPLE PRICING PERFORMANCE For esing he ou of sample pricing performance of he models, we calculae he model based opion prices for conracs raded a +, wih parameers ha were esimaed a ime in empirical mehodology secion, where are he arge observaion daes described above. hen, we calculae he pricing errors as he differences beween he forecased opion prices a ime and he marke opion prices known a ime +. hese ou of sample pricing errors are shown in Exhibi 3 below, in erms of RMSEs. hey follow a similar paern o he one repored for he in sample pricing resuls. In all cases, he GEV delivers smaller pricing errors han Black-Scholes. here is clearly some deerioraion in he ou of sample pricing performance for boh he Black-Scholes and he GEV opion pricing models. Again, he GEV model poss uniformly good performance over all he mauriy periods while Black-Scholes